Intersection of curves via iteration

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) where $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 4 x ^ { 2 } - 1 } & x > 0
\mathrm {~g} ( x ) = 8 \ln x & x > 0 \end{array}$$

1. Find
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { g } ^ { \prime } ( x )\) Given that \(\mathrm { f } ^ { \prime } ( x ) = \mathrm { g } ^ { \prime } ( x )\) at \(x = \alpha\)
2. show that \(\alpha\) satisfies the equation $$4 x ^ { 2 } + 2 \ln x - 1 = 0$$ The iterative formula $$x _ { n + 1 } = \sqrt { \frac { 1 - 2 \ln x _ { n } } { 4 } }$$ is used with \(x _ { 1 } = 0.6\) to find an approximate value for \(\alpha\)
3. Calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(\alpha\)

10. The function f is defined by \(\mathrm { f } ( x ) = \arccos x\) for \(0 \leq x \leq a\)
The curve with equation \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}

1. State the value of \(a\)
2. 1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
   2. Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$
3. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places. CONTINUE YOUR ANSWER HERE CONTINUE YOUR ANSWER HERE

5
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-05_444_757_548_251} The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 3 }\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { - 2 x }\). The diagram shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real root \(\alpha\).

1. Show that \(\alpha\) satisfies the equation \(\mathrm { h } ( x ) = 0\), where \(\mathrm { h } ( x ) = x ^ { 2 } + 3 - x \mathrm { e } ^ { 2 x }\).
2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } \left( 1 - 2 \mathrm { e } ^ { 2 x _ { n } } \right) - 3 } { 2 x _ { n } - \left( 1 + 2 x _ { n } \right) \mathrm { e } ^ { 2 x _ { n } } } .$$
3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. \section*{(d) In this question you must show detailed reasoning.} Find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = \frac { 2 } { 13 }\).

8

1. The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
   1. Express \(\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
   2. Use your answer to part (a)(i) to find the value of \(f ^ { \prime } ( 1 )\).
2. The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
   The graphs intersect at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\mathrm { f } ( x ) = 0\), where f is the function defined in part (a).
   2. Taking \(x _ { 1 } = 1\) as a first approximation to the root of the equation \(\mathrm { f } ( x ) = 0\), use the Newton-Raphson method to find a second approximation \(x _ { 2 }\) to the root.
      (3 marks)
3. The region enclosed by the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = 1\) and the \(x\)-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
   (3 marks)